A deterministic algorithm for sparse multivariate polynomial interpolation

Ben-Or, Michael; Tiwari, Prashant

doi:10.1145/62212.62241

Cited by 250 publications

(224 citation statements)

References 9 publications

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“…In complexity theory, identity testing played a major role in results such as IP = PSPACE [LFKN92,Sha92], MIP = NEXPTIME [BFL91], and the proof of the PCP theorem [AS98, ALM + 98]. Identity testing algorithms also lead to interpolation/learning algorithms for sparse polynomials, see [BT88,GKS90,KS01] and references within, for depth-3 circuits [Shp09,KS08], and for read-once arithmetic formulas [SV08, SV09].…”

Section: Polynomial Identity Testingmentioning

confidence: 99%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

293

275

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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions.The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.

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Section: Polynomial Identity Testingmentioning

confidence: 99%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

293

275

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“…Ben-Or/Tiwari [8] Zippel [49] KS [ Remark that Algorithm 6 given by Theorem 6 has a better total complexity and delay than KS algorithm when the degree is less or equal to 10. Open question: is it possible to turn Algorithm 6 into a fixed parameter algorithm?…”

Section: Resultsmentioning

confidence: 99%

“…with all possible monomials, can be done efficiently through inversion of a Vandermonde matrix. This method is always exponential in the number of variables, but better algorithms (both deterministic and probabilistic) have been designed for a polynomial represented either by a black box or a circuit [8,49,25,19]. Their complexity polynomially depends on the number of mono-mials of the polynomial which can be much smaller than the potential number of monomials.…”

Section: Motivations and Connection To Previous Workmentioning

confidence: 99%

On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation

Strozecki

2013

Theory Comput Syst

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We study the problem of generating the monomials of a black box polynomial in the context of enumeration complexity. We present three new randomized algorithms for restricted classes of polynomials with a polynomial or incremental delay, and the same global running time as the classical ones. We introduce TotalBPP, IncBPP and DelayBPP, which are probabilistic counterparts of the most common classes for enumeration problems. Our interpolation algorithms are applied to algebraic representations of several combinatorial enumeration problems, which are so proved to belong to the introduced complexity classes. In particular, the spanning hypertrees of a 3-uniform hypergraph can be enumerated with a polynomial delay. Finally, we study polynomials given by circuits and prove that we can derandomize the interpolation algorithms on classes of bounded-depth circuits. We also prove the hardness of some problems on polynomials of low degree and small circuit complexity, which suggests that our good interpolation algorithm for multilinear polynomials cannot be generalized to degree 2 polynomials. This article is an improved and extended version of [42] and of the third chapter of the author's phd thesis [41].

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“…This is an application of the Ben-Or/Tiwari algorithm [1] in the power basis of yj = xj + sj. Applying the early termination Ben-Or/Tiwari algorithm [12] to the power basis of yj = xj + sj, then when pj are distinct random values, without σ supplied as an input, the target polynomial can be interpolated in the given s-shifted basis with high probability.…”

Section: Sparse Interpolation On Shifted Bases With Early Terminationmentioning

confidence: 99%

“…Our algorithms are based on the early termination version [12] of the Ben-Or/ Tiwari sparse interpolation algorithm [1]. The main idea is that for a symbolic set of interpolation points, a shift must be a root of a discrepancy in the Berlekamp/Massey algorithm [18], which is called by the Ben-Or/Tiwari method.…”

Section: Introductionmentioning

confidence: 99%